The paper studies rigid sphere packings on 3D manifolds with boundary.
problem Investigating rigid sphere packings on 3D manifolds with boundary.
method Introducing generalized sphere packings, proving rigidity, introducing combinatorial curvature flows.
result Generalized sphere packing metrics are determined by combinatorial scalar curvature.
In this paper, we introduce two discrete curvature flows, which are called α-flows on two and three dimensional triangulated manifolds. For triangulated surface M, we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo \cite{CL1}), aiming at evolving α order di…
The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.
problem Understanding the combinatorial types of 3D polytopes in specific Riemannian manifolds.
method Analysis of mean curvature convex Riemannian polyhedra with non-obtuse dihedral angles in positive scalar curvature 3-manifolds.
result Determination of combinatorial types of 3D simple convex polytopes.
Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial for…
In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow metho…
Proves global rigidity of sphere packings on 3D manifolds.
problem Global rigidity of sphere packings on 3D manifolds.
method Combination of combinatorial scalar curvature and Andreev-Thurston rigidity theorem.
result Proves global rigidity of sphere packings on 3D manifolds.
Paper introduces new combinatorial curvature for surfaces with circle packing metrics.
problem Global rigidity of combinatorial curvature on triangulated surfaces.
method Innovative circle packing metrics, combinatorial Ricci flow, and constant curvature metrics.
result Existence of constant curvature metrics is equivalent to flow convergence.
In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulations of S3; namely, the boundary complexes of cyclic polyt…
Proves rigidity of sphere packings on 3D manifolds.
problem Rigidity of sphere packings on 3D manifolds.
method Combining combinatorial scalar curvature and Ricci curvature to prove rigidity.
result Proves infinitesimal rigidity of Thurston's Euclidean sphere packing.
The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.
In this work we construct a sequence of Riemannian metrics on the three-sphere with scalar curvature greater than or equal to 6 and arbitrarily large widths. Our procedure is based on the connected sum construction of positive scalar curvature metrics due to Gromov and Lawson. We develop analogies between the area of…
The paper studies rigidity of sphere packings on 3D manifolds with boundary.
problem Rigidity of sphere packings on 3D manifolds with boundary.
method Introduced generalized Thurston's sphere packings and proved their rigidity properties.
result Generalized Thurston's sphere packings are locally determined by combinatorial scalar curvatures and cannot be deformed while keeping combinatorial Ricci curvatures fixed.
The study examines 3D combinatorial flow in hyperbolic geometry, proving conditions for ball packings and convergence.
problem Analyzing 3D combinatorial Yamabe flow in hyperbolic geometry.
method Investigates triangulations and ball packings with vanishing combinatorial scalar curvature.
result Conditions for real or virtual ball packings and convergence of the flow.
The paper simplifies K-stability conditions for spherical varieties.
problem K-stability of polarized spherical varieties.
method Expressed K-stability in combinatorial terms, provided sufficient conditions.
result G-uniform K-stability provides a checkable condition for existence of constant scalar curvature metrics.
The paper proves a discrete positive mass theorem for graphs.
problem Formulating and proving a discrete positive mass theorem for graphs.
method Introducing asymptotically flat graphs, defining ADM mass, and using discrete harmonic functions.
result An asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph.
It has been pointed out to the author by David Glickenstein that the proof of the (closely related) Lemmas 1.2 and 3.2 in the title paper is incorrect. The statements of both Lemmas are correct, and the purpose of this note is to give a correct argument. The argument is of some interest in its own right.
Effective Yau-Tian-Donaldson conjecture for spherical varieties.
problem Finding effective K-stability criteria for spherical varieties.
method Formulated an effective variant of the Yau-Tian-Donaldson conjecture and reviewed effective K-stability criteria for spherical varieties.
result Effective K-stability criteria can be computed given combinatorial data.
The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
problem Finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
method Discrete uniformization theorem, combinatorial α-Yamabe flow, combinatorial α-Calabi flow, edge flipping surgery.
result Longtime existence and convergence of combinatorial α-Yamabe flow and combinatorial α-Calabi flow with surgery.
Study fundamental groups of small covers and their injective submanifolds.
problem Topology of small covers and their fundamental groups.
method Explicit presentations of fundamental groups and combinatorial data analysis.
result Characterization of 3D small covers with nonnegative scalar curvature.
Study of psc metrics via block diffeomorphisms and cubical sets.
problem Understanding the concordance of psc metrics.
method Constructing cubical sets and using block Dirac operators.
result Construction of a cubical Kan set and comparison map.
New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.
problem Finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows with surgery for inversive distance circle packings.
result Longtime existence and global convergence of combinatorial curvature flows with surgery.
Proves a conjecture for 3D Artin groups using new combinatorial curvature.
problem Proving the K(π,1) conjecture for Artin groups of dimension 3. method Introduces new combinatorial non-positive curvature.
result Proves the K(π,1) conjecture for Artin groups of dimension 3. The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.
problem Finding decorated piecewise hyperbolic metrics with prescribed combinatorial curvature.
method Introduced combinatorial α-Ricci flow with surgery to handle potential singularities and prove longtime existence and convergence.
result Existence of decorated piecewise hyperbolic metrics with prescribed combinatorial α-curvature.
Planar graphs' curvature total is a multiple of 1/12.
problem Total curvature of planar graphs with nonnegative combinatorial curvature.
method Proved using combinatorial curvature.
result Total curvature is an integral multiple of 1/12.
The paper examines complete Yamabe solitons with finite total scalar curvature.
problem Characterizing complete Yamabe solitons with specific curvature properties.
method Analyzing steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature.
result Steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.
The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
The paper studies special Finsler spaces with Hp-scalar curvature.
problem Characterizing and investigating Finsler spaces with specific scalar curvatures.
method Intrinsic investigation and various conditions for transformations between Finsler spaces.
result Conditions for transforming Finsler spaces of scalar curvature to those of Hp-scalar curvature. Study on Yamabe flow for negative scalar curvature.
problem Prescribed scalar curvature problem on compact manifolds.
method Yamabe flow with conditions on scalar curvature function.
result Long time existence and convergence of the flow.
The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n-volumic scalar curvature. method Using the curvature-dimension condition mCD(κ,n) and smGH-convergence. result The stability of n-volumic scalar curvature ≥κ under smGH-convergence. Extending results about positive scalar curvature to non-negative curvature.
problem Extending results about positive scalar curvature to non-negative curvature.
method Using index theory to generalize results.
result Explicit generalizations of classical results about positive scalar curvature metrics to non-negative curvature metrics.
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic S-curvature and are either Minkowskian or Riemannian. Paper introduces new flows to find circle packings with specific curvature.
problem Finding circle packings with prescribed total geodesic curvatures.
method Introduces combinatorial Calabi flow, fractional combinatorial Calabi flow, and combinatorial p-th Calabi flow.
result Establishes conditions for the longtime behaviors of these flows.
The study sets limits on scalar curvature in positive curvature manifolds.
problem Bounding scalar curvature in positive curvature manifolds.
method Establishing inequalities for manifolds with positive scalar curvature and scalar curvature bounded from below.
result Established inequalities for manifolds with positive scalar curvature.
Combinatorial approach to α-Ricci and Lin-Lu-Yau Ricci curvatures on graphs
problem Curvature formulas for α-Ricci and Lin-Lu-Yau Ricci curvatures on graphs method Combinatorial construction of optimal transport plans and exact formulas
result Combinatorial proof of known curvature formulas
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
Paper resolves spherical curvature flow problem.
problem Existence of ideal circle patterns in spherical background geometry.
method Introduces a combinatorial geodesic curvature flow in spherical background geometry.
result Characterizes sufficient and necessary conditions for flow convergence.
Study infinite combinatorial Ricci flow on spherical surfaces.
problem Investigate infinite combinatorial Ricci flow with spherical background.
method Establish existence and convergence of solution for infinite cellular decompositions.
result Existence and convergence of solution for infinite combinatorial Ricci flow in spherical geometry.
The paper studies Berwald scalar curvature properties in Finsler geometry.
problem Characterizing Finsler manifolds based on Berwald scalar curvature.
method Analyzes properties of Berwald scalar curvature and its implications for Finsler manifolds.
result Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
Small Weyl infimum on 4-manifolds with positive scalar curvature.
problem Understanding scalar curvature on 4-manifolds.
method Analyzing the Weyl functional and comparing scalar and self-dual Weyl curvatures.
result The infimum of the Weyl functional is small on many 4-manifolds with positive scalar curvature.
Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.
problem Stability and rigidity of scalar curvature in quaternion-Kähler manifolds.
method Analysis of stability and rigidity conditions using Einstein manifold properties.
result Quaternion-Kähler manifolds of negative scalar curvature are stable and scalar curvature rigid.
Proves curvature comparison for Riemannian bands in low dimensions.
problem Curvature comparison in Riemannian bands with lower bounds.
method Uses warped products over scalar-flat manifolds with log-concave warping.
result Scalar and mean curvature comparison results proven.
The paper defines Ricci curvature on cell-complexes and proves a Gauss-Bonnnet theorem.
problem Defining and proving geometric theorems on cell-complexes.
method Defining differential forms and Laplacian on cell-complexes, constructing Bochner-Weitzenböck formula, and calculating Ricci curvature combinatorially.
result Established a Gauss-Bonnnet theorem for cell-complexes.
The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.
problem Conditions for positive scalar curvature on manifolds with boundaries and their doubles.
method Analyzes the relationship between boundary conditions and positive scalar curvature metrics on manifolds and their doubles.
result Provides conditions for positive scalar curvature metrics on manifolds with boundaries and their doubles.
The study explores scalar curvatures on manifolds with boundary properties.
problem Understanding scalar curvatures on manifolds with boundary constraints.
method Presentation of problems and results related to scalar curvatures and mean curvatures of boundaries.
result Exploration of natural and artificial constructions in scalar curvature studies.
The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.